nLab
derived microlocalization

Contents

Idea

Microlocalization is a tool to study the propagation of singularities of solutions of partial differential systems, in order to study pullbacks of solutions of differential systems (which generalize products of distributions) and prove index theorems.

Derived microlocalization is an adaptation of the theory of microlocalization to the setting of derived global analytic geometry.

Motivation for a derived notion of microlocalization

In non-smooth situations, the usual normal and conormal bundle used in classical microlocalization is not well behaved, and one needs to take derived versions of them. Moreover, to study global analytic index theory, one needs a formulation of the theory of microlocalization in terms of derived loop stacks, in order to work out a cyclic and Hochshild version of index theorems (similar to the one developed by Kashiwara and Schapira in the microlocal formulation of index theory) that works on an arbitrary (e.g. integral) basis.

Loop space and deformation to the normal bundle: method I

We would like to define a family over (say) the disc D 1D^1 whose fiber at 00 is LX:=Hom(S 1,X)LX:=\Hom(S^1,X) and whose fiber at 11 is [LX/S 1][LX/S^1]. In the case of a usual scheme, this is done (by Vezzosi) by showing that LXLX identifies with Hom(BG a,X)\Hom(BG_a,X), and by making G aG_a act multiplicatively on the action of BG aBG_a on LXLX, to get the trivial action at 00 and the usual action at 11. This gives the desired family. Recall that BG aBG_a is the affinization of BS 1B\mathbb{Z}\cong S^1.

Loop space and deformation to the normal bundle: method II

Microlocalization in derived geometry involves the proper definition of the deformation to the normal bundle of a closed embedding YXY\subset X of global analytic spaces (or even stacks). One needs to consider a derived loop space approach to this construction because it allows to avoid the use of denominators in the definition of the Chern character (following Connes-Loday-Toen-Vezzosi) and in the development of more general global analytic index theory with integral coefficients.

The space of paths on XX based on YY is the groupoid acting on YY given by

P YX:={fHom(Δ 1,X),f(0)Y,f(1)Y}.P_Y X:=\{f\in \Hom(\Delta^1,X),\;f(0)\in Y,\;f(1)\in Y\}.

More concretely, this is given by the homotopy pullback

P YX Hom(Δ 1,X) ev 0×ev 1 Y×Y X×X \array{ P_Y X &\to& \Hom(\Delta^1,X) \\ \downarrow && \downarrow^{\mathrlap{ev_0\times ev_1}} \\ Y\times Y &\to& X\times X }

Notice that the natural projection P YXY×YP_Y X\to Y\times Y makes P YXP_Y X a groupoid (of paths in XX) acting on YY. In the case of the diagonal immersion Y=MM×M=XY=M\hookrightarrow M\times M=X, we get

P YXLM:=Hom(S 1,M).P_Y X\cong LM:=\Hom(S^1,M).

There is a natural projection p:P YXHom(Δ 1,X)Xp:P_Y X\to \Hom(\Delta^1,X)\sim X and P YXP_Y X is equiped with the natural structure of a groupoid acting on YY through the projection P YXY×YP_Y X\to Y\times Y. A more explicit description of the loop space (that is obtained by using the homotopy Δ 1Δ 0\Delta^1\sim \Delta^0) is given by the homotopy pullback

P YX=Y× X hY,P_Y X=Y\times^h_X Y,

which clearly has an interesting meaning only in the setting of derived geometry. See also at derived loop space and at Hochschild cohomology.

To make the above construction work for general Artin stacks, one needs to make it local for the smooth topology. This is done by replacing the loop space groupoid P YXP_Y X acting on YY by its formal completion along the identity morphism. This gives a formal groupoid P^ YX\hat{P}_Y X that is equivalent modulo the HKR theorem (in characteristic 00) to T Y[1]XT_Y[-1]X for a general Artin stack. Similarly, we will have L^M:=P^ M(M×M)T[1]M\hat{L}M:=\hat{P}_M(M\times M)\cong T[-1]M by HKR.

The deformation to the normal bundle in strict derived global analytic geometry is then simply given by the (false to be corrected) formula (with D 1=𝕄(R{X} )D^1=\mathbb{M}(R\{X\}^\dagger) for RR the base ind-Banach ring)

P YX˜:={fHom D 1(Δ 1×D 1,X×D 1),f(0,0)Y,f(1,0)Y,f(,1)(X\Y)f(x,t)(X\Y)t0}. \widetilde{P_Y X}:=\{f\in \Hom_{D^1}(\Delta^1\times D^1,X\times D^1),\;f(0,0)\in Y,\; f(1,0)\in Y,\;f(-,1)\in (X\backslash Y)\Rightarrow f(x,t)\in (X\backslash Y)\forall t\neq 0\}.

More concretely, this is given by the homotopy pullback (where U(1)=𝕄(R{X,Y} /(XY1))U(1)=\mathbb{M}(R\{X,Y\}^\dagger/(XY-1)) and we use the homotopy equivalence Hom(Δ 1,X)X\Hom(\Delta^1,X)\sim X)

P YX˜ Hom D 1(Δ 1×D 1,X×D 1) ev (0,0)×ev (1,0)×ev (,U(1))×ev (,1) Y×Y×Hom(Δ 1×U(1),(X\Y))×(X\Y) i×j×k X×X×Hom(Δ 1×U(1),X)×X \array{ \widetilde{P_Y X} &\to& \Hom_{D^1}(\Delta^1\times D^1,X\times D^1) \\ \downarrow && \downarrow^{\mathrlap{ev_{(0,0)}\times ev_{(1,0)}\times \ev_{(-,U(1))}\times \ev_{(-,1)}}} \\ Y\times Y\times \Hom(\Delta^1\times U(1),(X\backslash Y))\times (X\backslash Y) &\stackrel{i\times j\times k}{\to}& X\times X\times \Hom(\Delta^1\times U(1),X) \times X}

It has an evident natural projection t:P YX˜D 1t:\widetilde{P_Y X}\to D^1, and a natural projection P YX˜Y×Y\widetilde{P_Y X}\to Y\times Y that makes it a family of groupoids parametrized by D 1D^1 and acting on YY. There is also a natural projection p:P YX˜Xp:\widetilde{P_Y X}\to X given by ff(,1)f\mapsto f(-,1). We will denote s:P YXP YX˜s:P_Y X\to \widetilde{P_Y X} the fiber t 1(0)t^{-1}(0).

One may complete P YX˜\widetilde{P_Y X} along the unit of its groupoid structures, to get a family of formal groupoids P YX^\widehat{P_Y X} parametrized by D 1D^1, whose fiber at 00 will be the formal loop space P^ YX\hat{P}_Y X obtained by completing the path groupoid P YXP_Y X acting on YY along its identity morphism, and whose fiber at 11 will be the formal completion X^ Y\hat{X}_Y of XX along YY. There is a natural action of S 1S^1 on P YX^\widehat{P_Y X}. We will still denote t:P YX^D 1t:\widehat{P_Y X}\to D^1 the natural projection, s:P^ YXP YX^s:\widehat{P}_Y X\to \widehat{P_Y X} the fiber t 1(0)t^{-1}(0), and p:P YX^Xp:\widehat{P_Y X}\to X the natural projection (evaluation at (,1)(-,1)).

In the particular case of the diagonal embedding Y=MM×M=XY=M\hookrightarrow M\times M=X over a field of characteristic 00, the quotient of XX by this family of formal groupoids gives back (modulo a convenient HKR theorem) exactly Simpson’s non-abelian Hodge structure, that gives a family of formal stacks over D 1D^1 whose fiber at 00 is the tangent space TXTX (seen as the quotient of XX by the trivial action of T[1]XT[-1]X) and whose fiber at 11 is the so-called de Rham space X dRX_{dR} of XX.

A derived analog of microlocalization

We now work on an arbitrary derived Artin stack MM, such that the diagonal Y=MM×M=XY=M\to M\times M=X is an affine closed embedding.

The usual theory of microlocalization may be adapted to the derived global analytic setting by replacing the deformation to the normal bundle T YX˜𝔸 1\widetilde{T_Y X}\to \mathbb{A}^1 by the formal deformation to the normal bundle P YX^D 1\widehat{P_Y X}\to D^1.

The derived specialization functor is given on FD b(X)F\in D^b(X) by

ν Y(F):=s *p *FD b(P^ YX).\nu_Y(F):=s^*p^*F\in D^b(\hat{P}_Y X).

Remark that the loop space analog L *XL^*X of the odd cotangent bundle T *[1]XT^*[1]X (that should be dual to the odd tangent bundle T[1]XT[-1]X (related to LXLX by the HKR theorem) is given by S 1XS^1\otimes X (external tensor product by the simplicial circle). We will have for every stack YY, the canonical equivalence

Hom(S 1X,Y)Hom SSets(S 1,Hom dSt(X,Y)).\Hom(S^1\otimes X,Y)\cong \Hom_{SSets}(S^1,\Hom_{dSt}(X,Y)).

The derived Fourier-Sato transformation is given on FD b(P^ YX)F\in D^b(\hat{P}_Y X) by

Φ(F):=?.\Phi(F):=?.

The derived microlocalization functor is given on FD b(M)F\in D^b(M) by

μ(F):=Φ(ν Y(F))\mu(F):=\Phi(\nu_Y(F))

for Y=MM×M=XY=M\hookrightarrow M\times M=X.

A derived analytic analog of microlocalization

One may replace the simplicial circle S 1S^1, used in the definition of the derived loop space, by the unitary group U(1)U(1) of overconvergent global analytic geometry, to get a more analytic theory of microlocalization. In the complex situation, we will have U(1)S 1U(1)\cong S^1 up to D 1D^1-homotopy. Remark that the exponential map exp(i):S 1exp(i-):\mathbb{R}\to S^1 also has a meaning in overconvergent complex analytic geometry, if we see \mathbb{R}\subset \mathbb{C} as a closed subset equipped with its germs of analytic functions.

In the global analytic setting (without imposing homotopy invariance), there is no reason to have U(1)=D 1× ***U(1)=D^1\times_{*\coprod *} *. We thus prefer to use P 1:=D 1 U(1)D 1P^1:=D^1\coprod_{U(1)}D^1 as a natural global analytic parameter space for paths. We define

L Y X:={fHom(P 1,X),f(0)Y,f()Y}.L^\dagger_Y X:=\{f\in Hom(P^1,X), f(0)\in Y,\; f(\infty)\in Y\}.

Remark that up to D 1D^1_\mathbb{R}-homotopy, we get

L Y XL YX.L^\dagger_Y X\sim L_Y X.

We must now check that the natural morphism

References

Last revised on January 5, 2015 at 21:23:03. See the history of this page for a list of all contributions to it.